Let $F$ be an infinite field. Then is it true that there does not exist any surjection from $F$ onto $F[x]$?

Question

Let $F$ be an infinite field. Then is it true that $|F|<|F[x]|$, i.e., is it true that there does not exist any surjection from $F$ onto $F[x]$, i.e., there does not exist any injection from $F[x]$ into $F$ ?

Answer 1

The statement is wrong. Assume the field $F$ contains countably infinitely many elements. Then the polynomial ring $F[x]\simeq\bigoplus_{i=0}^{\infty}F\cdot x^i$ as an $F$-module. We know the countably infinite union of countably infinite sets is again countably infinite, then $|F[x]|=|F|$, i.e., there exists a surjection from $F$ to $F[x]$.

Hope this helps.

Answer 2

There certainly exists a surjection from $F$ onto $F[x]$. It seems likely that there is no surjective homomorphism...

In what sense? Well, for example, $F$ and $F[x]$ are both $F$-vector spaces; since $\dim(F)<\dim(F[x])$ there is no $F$-linear surjection.

Answer 3

There can even be algebraically nice maps between the two rings in the counterintuitive direction.

For example, let $F = \Bbb Q(x_1,x_2,\dots)$ be the field of rational functions in infinitely many variables over the rational numbers. Then there is an injective ring homomorphism $\Phi\colon F[x]\to F$, determined by $\Phi(x) = x_1$ and $\Phi(x_j) = x_{j+1}$ for $j\ge1$.